Curved exponential models have the property that the dimension of the minimal sufficient statistic is larger than the number of parameters in the model. Many econometric models share this feature. The first part of the paper shows that, in fact, econometric models with this property are necessarily curved exponential. A method for constructing an explicit set of minimal sufficient statistics, based on partial scores and likelihood ratios, is given. The difference in dimension between parameterand statistic and the curvature of these models have important consequences for inference. It is not the purpose of this paper to contribute significantly to the theory of curved exponential models, other than to show that the theory applies to many econometric models and to highlight some multivariate aspects. Using the methods developed in the first part, we show that demand systems, the single structural equation model, the seemingly unrelated regressions, and autoregressive models are all curved exponential models.
Log-Linear Models for Frequency Data: Sufficient Statistics and Likelihood Equations
